Topological quantum phase transition in an $S=2$ spin chain

We construct a model Hamiltonian for $S=2$ spin chain, where a variable parameter $\alpha$ is introduced. The edge spin is $S=1$ for $\alpha =0$ and $S=3/2$ for $\alpha =1$. Due to the topological distinction of the edge states, these two phases must be separated by one or several topological quantum phase transitions. We have studied the quantum phase transition by density matrix renormalization group calculation and proposed a phase diagram for this model.

Figure 1

(Color online) The sketch of spin-2 AKLT wave function. The circles represent sites and each spin-2 is decomposed into four spin-1/2 solid dots. Red lines connecting the dots stand for singlet states.

Figure 2

(Color online) (a) Ground-state energy per site $E_0/N$ and (b) the absolute value of its corresponding second derivative $\left|d^{2}E/d{\alpha }^2\right|$, as a function of $\alpha$, got by DMRG with 600 states and $\beta =1.0$ at different system sizes.

Figure 3

(Color online) The real-space spin-spin correlation functions $⟨{\mathbf{S}}_1\cdot {\mathbf{S}}_r⟩$ at different $\alpha$, obtained by the DMRG with $N=600$ and $\beta =1.0$.

Figure 4

(Color online) Dimer order parameters of the model Hamiltonian with different choices of $\alpha$ and $\beta$. (a) The existence of dimerization is established by choosing different system sizes $N$. (b) Dimer order parameters for different $\beta$. This parameter disappears for the critical $\beta =0.27$.

Figure 5

(Color online) Ground-state phase diagram of the model Hamiltonian got by DMRG with $N=600$. The location of second-order phase transition for different $\beta$ is indicated by the red line. The yellow region above red line belongs to SZH topological class while the light blue region belongs to AKLT class. Possible dimerization is shown in the green region sandwiched between these two phases.